The plural of deer is deer, but if you are referring to different species of deer, would it be correct to say "deers"?
Considering that the plural of fish is fish, but when referring to different species one would use fishes. Does this rule apply for other animals such as deer and sheep when referring to multiple species/breeds?
If not, any explanation as to why the difference would be appreciated.
Deers has been used that way.
From A Popular History of Mammalia, 1850:
The subgenus Muntjacus is distinguished from the other deers by the horns being supported on elongated pedicels ...
From Comparative Genomics, 2000:
In contrast to other deers, the muntjaks display chromosome numbers ranging from the lowest chromosome number known in mammals of 2n=6 (female) and 2n=7 (male Indian muntjak; Muntiacus muntjak vaginalis. Figure 8) to 2n = 46 (Chinese muntjak; Muntiacus reeversi).
I would say that either deers or deer would be acceptable for this type of usage. However, if you want to be indisputably correct, you could use species of deer. I also suspect this is the most common term for this meaning, as neither deer nor deers sounds quite correct to me here.
I'm afraid that "deers" is not correct, which is to say that the form has never been in common use in the sense you suggest, as far as the OED is aware.
As you say, it's common today to use "fish" as the plural of "fish" (e.g. "two fish"), and certain people insist that you must only use "fishes" when talking about different fish species. But the original form "fishes" was simply the plural of fish (e.g. "two fishes"), and didn't necessarily mean two species. A form ending in "s" occurs in the Old English Vespasian Psalter around 825 ("Fuglas heofenes & fiscas saes"), while the collective singular form "fish" used for the plural does not occur until the fourteenth century (OED s.v. 1b). The plural "fishes" is still frequently found alongside "fish", and the rule about using "fishes" to refer only to species of fish is probably a recent development to make sense of this redundancy. At any rate, the OED does not recognize a unique sense of fish meaning "species of fish", whose plural "fishes" distinguishes it from the more familiar noun with the plural "fish". It certainly is not usual in English that a word denoting an animal should have two alternate plurals, the one referring to multiple instances of one species, the other referring to multiple species.
You cannot use "deers" to mean "species of deer" because "deers" has never been a form in common use, whereas "fishes" has been and continues to be. For that reason "deers" grates on the ear of a native speaker in a way that "fishes" doesn't.
The form "deers" is occasionally (though very rarely) found: Hogg's Tales and Sketches (1817) speaks of a "place of rendezvous, to which the deers were to be driven." But none of the very few examples of "deers" given by the OED uses the form to indicate that species of deer are meant.
(Source: OED, s.v. "fish" n.1 and "deer" n.)
